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12 December 2016

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Bourne, D.P. and Conti, S. and Mueller, S. (2017) 'Energy bounds for a compressed elastic lm on asubstrate.', Journal of nonlinear science., 27 (2). pp. 453-494.

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https://doi.org/10.1007/s00332-016-9339-0

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Energy Bounds for a Compressed Elastic Film on a Substrate

David P. Bourne1, Sergio Conti2 and Stefan Muller2,3

22 December 2015

Abstract

We study pattern formation in a compressed elastic film which delaminates from a sub-strate. Our key tool is the determination of rigorous upper and lower bounds on the minimumvalue of a suitable energy functional. The energy consists of two parts, describing the twomain physical effects. The first part represents the elastic energy of the film, which is ap-proximated using the von Karman plate theory. The second part represents the fracture ordelamination energy, which is approximated using the Griffith model of fracture. A simplermodel containing the first term alone was previously studied with similar methods by sev-eral authors, assuming that the delaminated region is fixed. We include the fracture term,transforming the elastic minimization into a free-boundary problem, and opening the way forpatterns which result from the interplay of elasticity and delamination.

After rescaling, the energy depends on only two parameters: the rescaled film thickness,σ, and a measure of the bonding strength between the film and substrate, γ. We proveupper bounds on the minimum energy of the form σaγb and find that there are four differentparameter regimes corresponding to different values of a and b and to different folding patternsof the film. In some cases the upper bounds are attained by self-similar folding patterns asobserved in experiments. Moreover, for two of the four parameter regimes we prove matching,optimal lower bounds.

1 Introduction

Compressed elastic sheets such as plastic films and fabric often exhibit self-similar folding pat-terns. A typical example are folds in curtains, which decrease in number and increase in size fromtop to bottom as the folds merge. This coarsening phenomenon is also observed at the microscalein graphene and semiconductor films. The spontaneous delamination of prestrained semiconduc-tor films from their substrates produces blisters with rich, self-similar folding patterns [23]. Thisalso represents a challenge to manufacturers. Recently experimentalists have discovered how this,originally unwanted phenomenon, can be harnessed for thin film patterning and nanofabrication[25], for example to create nanotubes and nanochannels from prestrained semiconductor films [14],[31].

In the mathematical community there is an ongoing programme to understand why thesepatterns occur. The calculus of variations has proved to be a useful tool, where the patterns areviewed as minimisers of an elastic potential energy. This is the approach we take in this paper.

Motivated by the experiments of [31], we study a variational model of a two-layer materialconsisting of a rectangular elastic film on a substrate. The film is clamped along one edge to thesubstrate and is free on the other three sides. Due to a lattice mismatch between the film andthe substrate, the film suffers isotropic in-plane compression. It can relax this compression bydelaminating from the substrate and buckling, but a short-range attractive force between the filmand the substrate opposes this. We assign to the material the following energy:

(1.1) I(σ,γ) := IvK + IBo,

1Department of Mathematical Sciences, Durham University2Institute for Applied Mathematics, University of Bonn3Hausdorff Center for Mathematics, University of Bonn

1

where IvK is the von Karman energy of isotropically compressed plates:

(1.2) IvK[u , w, l1, l2] :=

∫ l1

0

∫ l2

0

|Du+DuT +Dw⊗Dw−Id|2 dx dy+(σl1)2

∫ l1

0

∫ l2

0

|D2w|2 dx dy

and IBo is a bonding energy that penalises delamination of the film from the substrate:

(1.3) IBo[u , w, l1, l2] := γ |(x, y) ∈ (0, l1)× (0, l2) : w(x, y) > 0|.

Here (0, l1)× (0, l2) is the set of material points of the film, l1 ≤ l2, and u(x, y) = (u(x, y), v(x, y))and w(x, y) are the in-plane and vertical displacements of the film from an isotropically compressedstate. The substrate is taken to be at height z = 0 so that film is bonded to the substrate atpoints where w = 0. The energy is rescaled so that (u, v, w) = (x/2, y/2, 0) corresponds to thestress-free, minimum energy state of the film, and (u, v, w) = (0, 0, 0) has positive energy andcorresponds to the isotropically compressed state. The energy has two parameters: 0 < σ < 1 isthe rescaled thickness of the film and γ ≥ 0 is a measure of attractive force between the film andthe substrate. We show how these are related to physical parameters such as the film thicknessand Young’s modulus in Appendix A, where the energy (1.1) is derived. The experimental setupis shown in Figure 1 and discussed below.

x

y

z

l1

l2

Figure 1: Geometry of the partially delaminated film. The bottom layer is the substrate, the middle layer isthe sacrificial buffer layer, which is removed by chemical etching in the region x > 0, and the top layer is thethin elastic film. The film is subject to compression at the boundary x = 0, where it is still attached to thebuffer layer. It may rebond to the substrate in the region x > 0. For simplicity, in our model we take thebuffer layer to have zero thickness. Consequently we refer to the material as a two-layer material rather thana three-layer material. The general case is discussed briefly in Appendix B.

We assume that the film is clamped to the substrate along the edge 0 × (0, l2):

(1.4) u(0, y) = 0, v(0, y) = 0, w(0, y) = 0, Dw(0, y) = 0

and is free on the other three sides of the rectangle (0, l1)× (0, l2). Since the film cannot go belowthe substrate we also have the positivity constraint

(1.5) w ≥ 0.

The von Karman energy (1.2) is the sum of a stretching energy IS, which penalises compressionand extension, and a bending energy IBe:

(1.6)

IS :=

∫ l1

0

∫ l2

0

|Du +DuT +Dw ⊗Dw − Id|2 dx dy,

IBe := (σl1)2

∫ l1

0

∫ l2

0

|D2w|2 dx dy.

The folding patterns in the experiments of [31], and in compressed thin films in general, canbe explained as the competition between the stretching and bending energies. The stretchingenergy has no minimum over the set of displacements satisfying the clamped boundary condition.

2

Its infimum is zero, and an infimising sequence can be constructed by taking a periodic foldingpattern, with folds perpendicular to the clamped boundary, and by sending the wavelength of thefolds to zero. This relaxes the compression in the film but sends the bending energy to infinity.The competition between the stretching and bending energies determines the scale of the pattern.

We shall see that for some range of the parameters (σ, γ) it is energetically favourable forthe film to form a self-similar folding pattern with the wavelength of the folds increasing awayfrom the clamped boundary. Close to the boundary a fine folding pattern is needed in orderto interpolate the folds to the clamped boundary condition without paying too much stretchingenergy. It would cost too much bending energy, however, to use such small folds in the wholedomain and so branching occurs to obtain larger folds in the bulk of the domain. The addition ofthe bonding energy, which is the novelty of this paper, complicates the situation further and givesa richer family of folding patterns.

Experimental Motivation. In the experiments of [31] they use a three-layer material, wherethe bottom layer is a substrate (e.g., Si), the middle layer is a buffer layer (e.g., SiO2), and thetop layer is a thin semiconductor film (e.g., SixGe1−x). Due to a lattice mismatch between thefilm and the buffer layer, the thin film suffers isotropic, in-plane compression. See Figure 1.

In the experiments, a slab of the buffer layer is removed by chemical etching. (An acid is usedto eat away the buffer layer from the side, without damaging the film or substrate. Once thedesired portion of the buffer layer has been removed the acid is washed out and the sample isdried.) This allows the thin film to partially relax the in-plane compression by folding. Since thebuffer layer is thin, it is observed that the folds of the film come into contact with the substrateand bond to it via attractive interfacial forces. In this way submicro and nano scale channels arefabricated, which can be used, e.g., in nanofluidic devices. The patterns formed by the channelsare self-similar, consisting of regions where the film is bonded to the substrate between folds thatbranch as they approach the boundary of the etched region.

The model above is a simplified model where we take the buffer layer to have zero thickness andtreat the material as a two-layer material, although many of our results extend to the case of thinbuffer layers as shown in Appendix B. Also, while our variational model is a major simplificationof the dynamic etching, rinsing and drying processes used in experiments, we still obtain goodqualitative agreement with experiments. More sophisticated models of delamination appear forexample in [3] and [12].

Main results. The type of self-similar folding patterns seen in experiments are difficult topredict. Typically the in-plane compression in the film is far past the critical buckling threshold andso standard buckling analysis (linear instability analysis) cannot predict the branching patterns.Minimising the energy I(σ,γ) numerically is also challenging since there are many local minimisers.Instead we construct approximate global minimisers of I(σ,γ) by hand. In the proof of Theorem2.1 we construct admissible displacement fields (u∗, w∗) satisfying upper bounds of the form

I(σ,γ)[u∗, w∗, l1, l2] ≤ c l1l2 σaγb

for some c, a, b > 0. The powers a and b depend on the region in the phase diagram; in Theorem2.1 we identify four parameter regimes A–D within the parameter space (σ, γ) ∈ (0, 1)× [0,∞)corresponding to different values of a and b and different folding patterns (u∗, w∗) of the thin film.These parameter regimes are shown in Fig. 2 and the values of a and b are given in Theorem 2.1.

We will see that the upper bound for regime A is attained when the film is bonded to thesubstrate everywhere, for regime B by a simple periodic folding pattern, and for regimes C andD by fold branching patterns. What distinguishes regimes C and D is that in regime C the filmis bonded to the substrate in large parts of the domain. Also regime C is bulk dominated in thesense that the order of the energy is determined by the deformation of the film in the interiorof the domain, whereas regime D is boundary dominated. Consequently we refer to regime Aas the flat regime, regime B as the laminate regime (borrowing a term from the theory of phase

3

transitions in metals), regime C as the localised branching regime, and regime D as the uniformbranching regime. Some of these patterns are shown in Figures 3–5.

For parameter regimes A and D we prove matching lower bounds:

min(u,w)

I(σ,γ)[u , w, l1, l2] ≥ c l1l2 σaγb

with the same a and b as in the upper bounds, but with a smaller multiplicative constant 0 < c < c.See Theorem 2.2. Therefore we obtain power laws of the form

min(u,w)

I(σ,γ)[u , w, l1, l2] ∼ l1l2 σaγb.

See Remark 2.3. This means that our approximate global minimisers (u∗, w∗), while not beingminimisers, do have the optimal energy scaling. Moreover we see that some of our optimal con-structions (u∗, w∗) exhibit self-similar folding patterns. In this sense we predict the patterns seenin experiments.

For regimes B and C it remains an open problem to prove matching upper and lower bounds.

Applications of the results. The upper bound constructions give scaling laws for the geometryof the patterns. For example, in parameter regime C the upper bound construction correspondsto a self-similar folding pattern with periodicity cells of the form shown in Figure 5, where theperiod 2h decreases towards the clamped boundary. The period 2h0, the fold width 2δ0 and thefold height α0 of the coarsest pattern (the folds furthest from the clamped boundary) satisfy thescaling laws

(1.7) h0 ∼ l1σ1/4γ1/16, δ0 ∼ l1σ3/4γ−5/16, α0 ∼ l1σ1/2γ−1/8.

These come from the proof of Proposition 6.6. These scaling laws can be written in terms of thefilm thickness, Young’s modulus, etc., using equation (A.7). Moreover, by comparing these scalinglaws with experiments, we could extract a value for the bonding strength γ, which is difficult tomeasure experimentally. The upper bound constructions could also be used as initial guesses fornumerical simulations.

Related work. The variational study of pattern formation in compressed thin films was initiatedby Ortiz and Gioia in the 90s [32, 22, 23], and has meanwhile attracted significant attention inthe mathematical literature. Our results generalise those of [10] and [27], who proved for the caseγ = 0 the optimal energy scaling min I(σ,0) ∼ σ. In [11] it is shown that the minimum energyscales the same way if the thin film is modelled using three-dimensional nonlinear elasticity, whichjustifies our choice of the von Karman approximation (see also [18]). The scaling is different,however, if the in-plane displacements of the film (u, v) are neglected, see [22, 32, 26]. The rigorousenergy scaling approach used in this paper was first used for the study of pattern formation inshape-memory alloys [28, 29] and has proven successful in the study of a variety of other patternformation problems, including for example confinement of elastic sheets and crumpling patterns[19], and the structure of flux tubes in type-I superconductors [17, 16]. The limit in which thevolume fraction of one phase is very small may lead to the occurrence of a variety of phases withpartial branching, as was demonstrated for superconductors in [17, 16, 20] and for shape-memoryalloys and dislocation structures in [35, 21]. A finer mathematical analysis was possible in thecase of annular thin films [8]. A related problem, but without boundary conditions, has beenstudied in the case of films attached to a compliant substrates, and leads to different patterns,which are homogeneous over the film, see for example [30, 7] and references therein. There is alarge literature on folding patterns in compressed thin films and other approaches include linearinstability analysis, post-buckling analysis and numerical methods, e.g., [4, 5, 6, 14, 24, 33]. Thesetype of techniques were used by [3] to study the same experiments [31] as we do. Our resultscomplement theirs; they use a different model and techniques to obtain different types of results,namely quantitative predictions about the folding patterns away from the boundary, rather than

4

focussing on self-similar branching as we do here. In the physics literature self-similar foldingpatterns are referred to as wrinklons [34], see also [9] for a variational analysis. A more detaileddiscussion of the literature and an overview of the present results are given in the companion paper[13].

Outline of the paper. In Section 2 we state our main results, the upper and lower bounds onthe minimum value of the energy I(σ,γ). The lower bounds are proved in Section 3 and the upperbounds are proved in Sections 4–6. The model is derived in Appendix A. In Appendix B we showhow our results can be extended to include the more general boundary conditions that correspondto the the experiments of [31]. In Appendix C we prove a version of the Poincare inequality thatis needed for the lower bounds.

Notation. We denote by IvK[u , w, l1, l2] the functional defined in (1.2), and occasionally omitthe parameters l1 and l2 for brevity. We use IvK[u , w, ω] for the same functional with integrationdomain restricted to ω, and the same for the indiviudual components IS, IBe, IBe, IBo; we omitthe arguments if they are clear from the context. Let Ω := (0, l1) × (0, l2) be the set of materialpoints of the elastic film, 0 < l1 ≤ l2, and let

(1.8) V :=

(u , w) ∈ H1(Ω;R2)×H2(Ω) : w = 0, u = Dw = 0 on 0 × (0, l2), w ≥ 0

be the set of admissible displacements satisfying the clamped boundary condition and the positivityconstraint.

We use standard notation for the Sobolev spaces W 1,p(Ω;Rm) and their traces. We useSBV (Ω;Rm) for the space of special functions of bounded variation, defined as the set of f ∈L1(Ω;Rm) such that their distributional derivative Du is a bounded measure, given by the sum ofa part absolutely continuous with respect to the Lebesgue measure and a part concentrated on a(one-rectifiable) jump set Ju , see for example [2] for the standard notation and properties. We useSBD(Ω) for the space of functions of bounded deformation, for which only the symmetric partof the distributional derivative has the stated property, see [1, 15] for the relevant properties. Wewrite H1 for the one-dimensional Hausdorff measure. We denote by c a generic positive constantthat may change from appearence to appearence, but does not depend on the parameters of theproblem.

2 Main Results

In this section we state our main results. The proofs will be postponed to the following sections.

Theorem 2.1 (Upper bounds). Let 0 < l1 ≤ l2. Define the parameter space X = (σ, γ) ∈(0, 1)× [0,∞). Define parameter regimes

A :=(σ, γ) ∈ X : γ ≥ σ−1,B :=(σ, γ) ∈ X : σ−4/9 ≤ γ < σ−1,C :=(σ, γ) ∈ X : σ4/5 ≤ γ < σ−4/9,D :=(σ, γ) ∈ X : γ < σ4/5,

see Figure 2. There exists a positive constant c, independent of l1, l2, σ, and γ, such that

minV

I(σ,γ) ≤ c l1l2

1 if (σ, γ) ∈ A,(σγ)2/5 if (σ, γ) ∈ B,σ1/2γ5/8 if (σ, γ) ∈ C,σ if (σ, γ) ∈ D.

Proof. The proof is done in Sections 4–6, separately for each regime. Precisely, the bound inRegime A follows from Lemma 4.1, the one in regime B from Proposition 5.3, regime C and Dfrom Proposition 6.6.

5

1

100

A

B

C

D

σ

γ

σγ = 1

γ = σ−4/9

γ = σ4/5

Figure 2: The four parameter regimes given in Theorem 2.1. A is the flat regime, B is the laminate regime,C is the localised branching regime, and D is the uniform branching regime.

Define an additional parameter regime

D′ := (σ, γ) ∈ X : γ < σ1/2 ⊃ D.

Theorem 2.2 (Lower Bounds). Let γ ≥ 0, σ ∈ (0, 1), 0 < l1 ≤ l2. There exists a positive constantc, independent of l1, l2, σ, and γ, such that for all (u , w) ∈ V

(2.1) I(σ,γ)[u , w, l1, l2] ≥ c l1l2

1 if γ ≥ σ−1 (regime A),(σγ)2/3 if σ1/2 ≤ γ ≤ σ−1,σ if γ < σ1/2. (regime D′).

Proof. This follows immediately from Lemma 3.1 and Lemma 3.2 in Section 3 below, using thefact that (σγ)2/3 ≥ σ is equivalent to γ ≥ σ1/2.

Remark 2.3 (Optimality of the bounds in regimes A and D). The bounds for regimes A and Dare optimal in the sense that the lower and upper bounds scale the same way in the parameters σand γ:

minV

I(σ,γ) ∼

1 if (σ, γ) ∈ A,σ if (σ, γ) ∈ D.

Proving optimal bounds in the whole parameter space X remains an open problem.

Idea of the proof of Theorem 2.1. We describe the constructions used to obtain the upperbounds in Theorem 2.1, which correspond to different folding patterns of the thin film. The typeof pattern is determined by the competition between the stretching, bending and bonding energies.

In the flat regime A the bonding strength γ is large compared to σ and the upper boundI(σ,γ) ≤ c l1l2 is obtained by taking w = 0 everywhere, i.e., by taking the film to be bonded tothe substrate everywhere. See Lemma 4.1. In regimes B–D, where the bonding strength γ issmaller, this upper bound can be improved by allowing the film to relax the in-plane compressionby partially delaminating from the substrate and buckling.

6

In the laminate regime B we obtain the upper bound I(σ,γ) ≤ c l1l2(σγ)2/5 with a simpleperiodic folding pattern interpolated to the clamped boundary conditions, as shown in Figure 3.Note that the film is still bonded to the substrate in a large part of the domain. See Proposition5.3. In regimes C and D a better upper bound can be achieved using a self-similar branchingpattern, where the period of the folding pattern decreases towards the clamped boundary.

Figure 3: The upper bound construction for the laminate regime B. Away from the clamped boundary thefilm forms a periodic pattern consisting of folds in between regions where it is bonded to the substrate. Thisis interpolated to the clamped boundary conditions in a thin boundary layer.

In the subset of regime D where γ ≤ σ the bonding energy is a lower order term and theupper bound I(σ,γ) ≤ c l1l2σ can be obtained using the same period-halving construction that wasused for the case γ = 0 in [10], as shown in Figure 4. Note that the film is delaminated from thesubstrate almost everywhere.

Figure 4: The upper bound construction for parameter regime γ ≤ σ ⊂ D. Away from the clampedboundary the film forms a periodic folding pattern (right). The folds undergo a sequence of period-halvings(centre) before being interpolated to the clamped boundary (left). The film is delaminated from the substratealmost everywhere, but we do not pay too much bonding energy for this since the bonding strength γ is smallcompared to σ.

In the rest of regime D and in regime C a new upper bound construction is needed. Awayfrom the clamped boundary we take the film to form a periodic pattern, with periodicity cells ofthe form shown in Figure 5. It is energetically favourable for the period of the pattern to decreasetowards the clamped boundary. By construction the stretching energy of each periodicity cell iszero. By requiring that the bending energy of each periodicity cell equals its bonding energy,we find that fold width 2δ is related to the period 2h by δ ∼ h1/3. Therefore, as we halve theperiod, h 7→ 2−1h, we must multiply the fold width by a factor of 2−1/3 ≈ 0.8. This means thatthe relative area δ

h where the film is delaminated in the periodicity cell increases as h decreases.In order to achieve this we perform each period halving h 7→ 2−1h using a two step refinementprocedure consisting of fold shrinkage (Figure 6, right) and fold splitting and separation (Figure6, left). (For a real film the two branching steps would happen simultaneously, but this does notchange the order of the energy.) This takes one cell of period 2h and fold width 2δ and producestwo cells of period h and fold width 22/3δ ≈ 1.6δ.

We repeat this refinement process until the pattern is fine enough to be interpolated to theclamped boundary conditions without paying too much stretching energy. We will see in the proof

7

of Proposition 6.6 that if γ ≥ 1, then refinement stops before δ = h, i.e., before the film becomesdelaminated from the substrate almost everywhere.

2δ2h

Figure 5: One of the periodicity cells used for regimes B, C, D. A fold of width 2δ lies between two regionswhere the film is bonded to the substrate.

Figure 6: The branching construction used for regimes C and D. Each refinement consists of two steps: foldshrinkage (right), where a fold is reduced in width and height, and fold splitting (left), where a fold splits intotwo folds of the same width.

If γ < 1, however, the pattern refinement continues until δ = h and the film is delaminatedfrom the substrate almost everywhere. At this point the period of the folds is still too large;if refinement were stopped here then the stretching energy in the interpolation boundary layerwould be too high. Therefore we continue refining the pattern towards the clamped boundary byperiod-halving, δ 7→ 2−1δ, using the same construction as for regime γ ≤ σ, shown in Figure 4.This continues until δ = l1σ at which point the pattern is fine enough to be interpolated to theclamped boundary conditions. See the proof of Proposition 6.6.

3 Lower Bound Proofs

In this section we prove the main ingredients of the lower bound in Theorem 2.2.

Lemma 3.1 (Lower bound for γ ≥ 0). Let γ ≥ 0, σ ∈ (0, 1], 0 < l1 ≤ l2. Then for all (u , w) ∈ V

I(σ,γ)[u , w, l1, l2] ≥ c l1l2σ.

Proof. By scaling it suffices to consider the case l1 = 1. Further, it suffices to prove the boundfor l2 = 1, and apply it to each of the squares (0, 1)× (k, k + 1), k ∈ N ∩ [0, l2 − 1], and use thatbl2c ≥ l2/2 for all l2 ≥ 1.

The bound for l1 = l2 = 1 and γ = 0 was proven in [10, Lemma 1] and implies the general caseγ ≥ 0 since I(σ,γ) ≥ I(σ,0).

8

Lemma 3.2 (Lower bounds for γ ≥ σ2). Let σ ∈ (0, 1], γ ≥ σ2. Then for all (u , w) ∈ V

I(σ,γ)[u , w, l1, l2] ≥ c l1l2

1 if γ ≥ σ−1,(σγ)2/3 if σ2 ≤ γ ≤ σ−1.

Proof. As above, it suffices to consider the case l1 = l2 = 1. Let l ∈ (0, 1] with 1/l an integer.Subdivide Ω := (0, 1)2 into l−1 squares of side length l, denoted by qij :

qij := l(i, i+ 1)× l(j, j + 1), i, j ∈ 0, . . . , l−1 − 1.

We say that qij is good if

|w = 0 ∩ qij | ≥l2

2.

Otherwise qij is bad. Let NB be the number of bad squares, ΩG be the union of all the goodsquares, and ΩB be the union of the bad ones. Let E be the energy of a given deformation (u , w).It is easy to see that

NBl2

2≤ |ΩB ∩ w > 0| ≤ |w > 0| ≤ E

γ,

which implies

(3.1) NB ≤2E

γl2.

Let qij be a good square. By the Poincare inequality

(3.2)

∫qij

|Dw|2 dx ≤ l2∫qij

|D2w|2 dx .

(The Poincare constant l2 is obtained in Appendix C.) Summing over all good squares yields

(3.3)

∫ΩG

|Dw|2 dx ≤ l2∫

ΩG

|D2w|2 dx ≤ l2

σ2E.

Define

η :=

∫ΩG

|Du +DuT − Id| dx .

We claim that

(3.4) η ≤ E1/2 +l2

σ2E.

Proof: First note that

(3.5) E ≥∫

Ω

|Du +DuT − Id +Dw ⊗Dw|2 dx ≥(∫

Ω

|Du +DuT − Id +Dw ⊗Dw| dx)2

by the Cauchy-Schwarz inequality since Ω has area 1. By the definition of the Frobenius norm

|Dw ⊗Dw|2 = w4x + 2w2

xw2y + w4

y = (w2x + w2

y)2 = |Dw|4.

Therefore

(3.6)

∫ΩG

|Dw ⊗Dw| dx =

∫ΩG

|Dw|2 dx ≤ l2

σ2E

by (3.3). Using the triangle inequality and combining (3.5) and (3.6) gives

η ≤∫

ΩG

|Du +DuT − Id +Dw ⊗Dw| dx +

∫ΩG

|Dw ⊗Dw| dx ≤ E1/2 +l2

σ2E

9

x

y

Figure 7: Sketch of the geometry in the proof of Lemma 3.2. The squares in ΩB are shown, as well as someof the lines (0, y) + nR.

as required.At this point we define U (x ) := (2u(x )−x )χΩG

(x ). Then U ∈ SBV (Ω;R2) ⊂ SBD(Ω) andits jump set JU is contained in the union of the boundaries of the bad squares, hence H1(JU ) ≤4lNB ≤ 8E/γl by (3.1). At the same time, ‖e(U )‖L1(Ω) = η (see the end of the Introduction forthe notation). By the trace part of the Korn-Poincare inequality for SBD functions proven in [15,Th. 1] (with p = 1) we obtain that there are a set ωΓ ⊂ ∂Ω with H1(ωΓ) ≤ cH1(JU ) and an affinefunction a with Da +DaT = 0 such that

∫0×(0,1)\ωΓ

|a(x )− x |dH1(x ) ≤ cη for some universal

constant c > 0. Since a2 is constant on this set, if the length of ωΓ is less than 1/2 we obtainη ≥ c for some (different) universal constant c. Conversely, if the length of ωΓ is more than 1/2we obtain 1/2 ≤ cH1(JU ) ≤ cE/γl. Therefore at least one of E ≥ cγl and η ≥ c holds. Recalling(3.4) this gives

(3.7) E ≥ cmin

γl, 1,

σ2

l2

for all l ∈ (0, 1] with l−1 ∈ N. It then follows that the same estimate (with a different constant)holds for all l ∈ (0, 1].

To conclude the proof we consider different choices of l. Equating the second two terms on theright-hand side of (3.7) gives l = σ and

E ≥ cminσγ, 1 = c if σγ ≥ 1.

Equating the first and third terms on the right-hand side of (3.7) gives l = σ2/3γ−1/3 and

E ≥ cmin(σγ)2/3, 1 = c(σγ)2/3 if σγ ≤ 1.

Since we require that l ∈ (0, 1], then this estimate is only valid if in addition σ2/3γ−1/3 ≤ 1 ⇐⇒γ ≥ σ2. This completes the proof.

In order to make the argument more transparent we also give a self-contained proof, whichdoes not apply the rigidity estimate in [15, Th. 1] but instead makes direct usage of some ideasfrom its proof, which can be much simplified in the present context.

We assume for now that

(3.8) E ≤ 1

32γl.

Therefore by inequalities (3.1) and (3.8)

(3.9) NBl ≤1

16.

Pick n := (cos θ, sin θ) ∈ S1 with n1 ≥ 8|n2|. Define

ωn :=y ∈ (0, 1) : (0, y) + nR ∩ ΩB = ∅

,

10

see Figure 7. Note that L1(ωcn) is maximised, e.g., when n1 = 8n2 and the union of bad squaresΩB is given by

NB⋃i=1

(0, l)× l( i8 + (i− 1), i8 + i).

In this case each line (0, y) + nR intersects at most one bad square and L1(ωcn) = 98 lNB ≤2lNB .

By combining this and (3.9) we obtain

L1(ωn)= 1− L1(ωcn) ≥ 1− 2lNB ≥7

8

for all admissible n . For y ∈ ωn let In(y) := (0, y) + nR ∩ (0, 1)2 and

Ωn :=⋃y∈ωn

In(y).

A simple geometrical argument shows that |Ωn | ≥ 13/16 for all n such that n1 ≥ 8|n2|, so that|Ωcn | ≤ 3/16.

We estimate

(3.10)

∫Ωn

|Du +DuT − Id| dx ≤∫

ΩG

|Du +DuT − Id| dx = η.

The function fn(x ) := 2n · u(x )− xn1

vanishes at x = 0 since u satisfies the boundary conditionu(0, y) = (0, 0). Also

(3.11) Dfn · n = 2n ·Du · n − 1 = n · (Du +DuT − Id)n .

By the Poincare inequality on each line, (3.11) and (3.10)

(3.12)

∫Ωn

|fn | dx ≤ c∫

Ωn

|Dfn · n | dx ≤ c∫

Ωn

|Du +DuT − Id| dx ≤ cη

for some c > 0. Fix three different, admissible values of n , called n , n ′, n ′′, with correspondingangles θ, θ′, θ′′. Set Ω = Ωn ∩ Ωn′ ∩ Ωn′′ . Then

|Ω|= 1− |Ωc| = 1− |Ωcn ∪ Ωcn′ ∪ Ωcn′′ | ≥ 1− (|Ωcn |+ |Ωcn′ |+ |Ωcn′′ |) ≥ 1− 3 · 3

16=

7

16.

Given x > 0, consider the following over-determined linear system for v ∈ R2:

2n · v =x

n1, 2n ′ · v =

x

n′1, 2n ′′ · v =

x

n′′1,

which can be written as

(3.13) 2

nn ′

n ′′

v = x

1/n1

1/n′11/n′′1

if we consider n , n ′, n ′′ to be row vectors. It is easy to check that the null-space of the adjointof the matrix on the left-hand side is spanned by (sin(θ′ − θ′′), sin(θ′′ − θ′), sin(θ− θ′)). Therefore(3.13) has a solution v if and only if

0 =

1/n1

1/n′11/n′′1

·sin(θ′ − θ′′)

sin(θ′′ − θ′)sin(θ − θ′)

=

1

2

(cos2 θ(sin 2θ′′ − sin 2θ′) + cos2 θ′(sin 2θ − sin 2θ′′) + cos2 θ′′(sin 2θ′ − sin 2θ)

).

11

Now choose θ > 0, θ′ = −θ. Then the right-hand side is nonzero since θ′′ 6= ±θ, and there is nosolution v to (3.13). Therefore for this choice of n , n ′, n ′′

(3.14) minv∈R2

(∣∣∣∣2n · v − x

n1

∣∣∣∣+

∣∣∣∣2n ′ · v − x

n′1

∣∣∣∣+

∣∣∣∣2n ′′ · v − x

n′′1

∣∣∣∣) > 0

for any x > 0 and uniformly if x ≥ x∗, for some x∗ > 0. By (3.4), (3.12), (3.14)

E1/2 +l2

σ2E ≥ η

≥ 1

3c

∫Ω∩x≥x∗

|fn |+ |fn′ |+ |fn′′ | dx

≥ 1

3c|Ω ∩ x ≥ x∗| min

v∈R2

x∈[x∗,1]

(∣∣∣∣2n · v − x

n1

∣∣∣∣+

∣∣∣∣2n ′ · v − x

n′1

∣∣∣∣+

∣∣∣∣2n ′′ · v − x

n′′1

∣∣∣∣)≥ c(3.15)

for some constant c > 0. Putting together (3.8) and (3.15) leads to (3.7), and the proof is concludedas above.

4 Upper Bound Construction for the Flat Regime A

In this section we prove Theorem 2.1 for parameter regime A. In this regime γ is very largeand the upper bound is obtain by taking the film to be bonded to the substrate everywhere.

Lemma 4.1 (Energy of the flat construction). There exists a constant c > 0 such that for allσ ∈ (0, 1), γ > 0

(4.1) minV

I(σ,γ) ≤ c l1l2.

Proof. The displacement field (u, v, w) = (0, 0, 0) satisfies the desired upper bound (4.1). Notethat a smarter choice of displacement field is (u, v, w) = (x/2, 0, 0), which yields the same bound(4.1) but with a smaller constant c. Here the film relaxes compression in the x-direction, thedirection orthogonal to the clamped boundary, by spreading out.

5 Upper Bound Construction for the Laminate Regime B

In this section we prove Theorem 2.1 for parameter regime B. In this regime it is favourablefor the film to buckle, but not to exhibit branching patterns. See Figure 3.

In this and the next Section we work in components, so we write x = (x, y) and u = (u, v),the stretching energy takes the form

(5.1) IS =

∫Ω

(2ux + w2x − 1)2 dx + 2

∫Ω

(uy + vx + wxwy)2 dx +

∫Ω

(2vy + w2y − 1)2 dx .

5.1 Construction in the Interior

First we construct a displacement field on an interior rectangle of (0, l1) × (0, l2), away fromthe clamped boundary x = 0. By translation invariance of the energy we can take the rectangleto be (0, l)× (−h, h) with l < l1, 2h ≤ l2. Take δ < h and consider a vertical displacement of theform

(5.2) w(x, y) := w(y) :=

0 y ∈ (−h,−δ],

α

2

(1 + cos

πy

δ

)y ∈ [−δ, δ],

0 y ∈ [δ, h).

12

This describes an x-independent displacement where the film has one fold of height α ≥ 0 andwidth 2δ, and is bonded down to the substrate elsewhere, see Figure 5. Choosing u and v so thatthe stretching energy vanishes, i.e., IS = 0, and with v(x,−h) = v(x, h) = 0 yields

(5.3)

u(x, y) := u(x) :=1

2x+ d,

v(x, y) := v(y) :=

1

2(y + h) y ∈ (−h,−δ],

y

2− α2π2

16δ2

(y − δ

2πsin

2πy

δ

)y ∈ [−δ, δ],

1

2(y − h) y ∈ [δ, h),

for any constant d. In order for v to be continuous we must have

(5.4) α2 =8

π2δh.

We remark that the boundary conditions v(x,−h) = v(x, h) = 0 are imposed in order to be ableto extend v periodically in the y direction; since v is affine for |y| > δ the total compression in(−h, h) is accommodated in the central region (−δ, δ). This is the physical origin of the value ofα given in (5.4). It is a simple calculus exercise to verify the following:

Lemma 5.1 (Energy of the basic laminate construction). The displacement (u, v, w) defined inequations (5.2)–(5.4) has the following energy on the rectangle (0, l)× (−h, h):

I(σ,γ) = 2π2(σl1)2 lh

δ2+ 2γlδ.

5.2 Construction in the Boundary Layer

We define a displacement field (u, v, w) on a boundary layer rectangle (0, ε) × (−h, h) byinterpolating between the clamped boundary conditions (1.4) and the displacements introducedin equations (5.2)–(5.4): Let

(5.5) u(x) :=1

2x, v(x, y) := ψ2

(xε

)v(y), w(x, y) := ψ

(xε

)w(y),

where ψ is the cubic interpolating polynomial satisfying ψ(0) = 0, ψ(1) = 1, ψ′(0) = ψ′(1) = 0,i.e.,

ψ(t) := t2(3− 2t).

Note that in (5.5) we take v = ψ2v rather than the more natural choice of v = ψv since then thestretching term w2

y + 2vy − 1 is of order 1 rather than of order h/δ. See equation (5.9).We extend (u, v, w) to the whole boundary layer (0, ε)× (0, l2) by gluing together l2/2h copies

of (u, v, w) along their horizontal boundaries. (If 2h does not divide l2, then we glue togetherbl2/2hc copies to get a construction on (0, ε) × (0, 2hbl2/2hc) and define u = x/2, v = w = 0 on(0, ε)× (2hbl2/2hc, l2). For simplicity we assume that 2h divides l2 in the rest of the paper sinceit does not affect the energy bound.)

Lemma 5.2 (Energy of the boundary layer construction). Let 0 < δ ≤ ε, δ ≤ h ≤ l2. Thedisplacement field (u, v, w) : (0, ε)× (0, l2)→ R3 defined above satisfies the following energy bound:

(5.6) I(σ,γ)[u, v, w, ε, l2] ≤ cl2(h2

ε+ ε+ (σl1)2 ε

δ2+ γ

εδ

h

).

Proof. Observe that w, defined in equation (5.2), satisfies the following:

|∂my w| ≤ ch1/2δ1/2−m for m ≥ 0.

13

Thus w satisfies

(5.7) |∂nx∂my w| ≤ ch1/2δ1/2−mε−n for m ≥ 0, n ≥ 0.

The displacement v, defined in equation (5.3), satisfies |v| ≤ ch and so

(5.8) |vx| ≤ ch

ε.

Recall that v was chosen so that w2y + 2vy − 1 = 0. Therefore

(5.9) |w2y + 2vy − 1| = |ψ2(w2

y + 2vy − 1)− (1− ψ2)| = |1− ψ2| ≤ 1.

From equations (5.7)–(5.9) we can estimate the stretching, bending, and bonding energy in theboundary layer (0, ε)× (0, l2):

IS[u, v, w, ε, l2] ≤ cl2(hδ3

ε3+h2

ε+ ε

), IBe[u, v, w, ε, l2] ≤ c l2

h(σl1)2εδ

(hδ

ε4+

h

δε2+h

δ3

),

(5.10)

IBo[u, v, w, ε, l2] ≤ c l2hγεδ.

By using the assumptions δ ≤ ε and δ ≤ h, the bounds (5.10) reduce to (5.6), as required.

5.3 Complete Construction

We are now in a position to prove the upper bound for the laminate regime B:

Proposition 5.3 (Energy of the laminate construction). Let 0 < l1 ≤ l2, σγ ≤ 1, γ ≥ 1. Then

minV

I(σ,γ) ≤ cl1l2(σγ)2/5.

Proof. Take the displacement field (u, v, w) that was defined in (5.2)–(5.4) on (0, l)× (−h, h) andextend it to the domain U =(ε, l1)× (0, l2) by taking l = l1−ε, d = ε/2, and gluing together l2/2hcopies along their horizontal boundaries (assuming without loss of generality that 2h divides l2 asabove). One can easily check that (u, v, w,Dw) match continuously at the boundaries. Lemma5.1 implies that this construction has energy

(5.11) I(σ,γ)[u, v, w, U ] ≤ c (l1 − ε)l2h

((σl1)2 h

δ2+ γδ

).

Define (u, v, w) on (0, ε)×(0, l2) using the boundary layer construction from Section 5.2. As above,(u, v, w,Dw) match continuously at the boundaries. By combining (5.6) and (5.11) we find thatthe energy on the whole domain (0, l1)× (0, l2) satisfies

(5.12)

I(σ,γ)[u, v, w, l1, l2] ≤ cl2(h2

ε+ ε+ (σl1)2 ε

δ2+ γ

εδ

h+ (σl1)2 (l1 − ε)

δ2+ γ

(l1 − ε)δh

)= cl2

(h2

ε+ ε+ (σl1)2 l1

δ2+ γ

l1δ

h

).

Now we chose ε, h, and δ to minimise the order of the energy. This can be done by equating termson the right-hand side of (5.12), which yields

(5.13) h = l1(σγ)2/5, ε = h, δ = (σl1)2/3γ−1/3h1/3 = l1σ4/5γ−1/5.

Note that the conditions l1 ≤ l2, σγ ≤ 1, γ ≥ 1 ensure that the geometrical restrictions δ ≤ h,h ≤ l2, ε ≤ l1 and the constraint δ ≤ ε are satisfied. Substituting (5.13) into (5.12) gives

I(σ,γ)[u, v, w, l1, l2] ≤ l1l2(σγ)2/5

as required.

14

6 Upper Bound Constructions for the Branching RegimesC and D

In this section we prove Theorem 2.1 for the branching regimes C and D. The basic idea is touse copies of a folding pattern of the form shown in Figure 5 and to decrease h and δ towards theclamped boundary by a sequence of fold branchings.

6.1 Branching constructions.

First we prove a general lemma about the cost of fold branching. This covers both our foldsplitting construction (Lemma 6.3) and our fold shrinkage construction (Lemma 6.4).

Lemma 6.1 (The cost of branching). Let ω := (0, l)× (−h, h), h ≤ l. Let 0 < δ ≤ h, c∗ > 0, andlet w ∈ C4(ω; [0,∞)) satisfy the following:

(i) w(x, y) = 0 for y in a neighbourhood of ±h; wx(x, y) = 0 for x in a neighbourhood of 0 andl;

(ii) w(x,−y) = w(x, y);

(iii) For all x ∈ (0, l) ∫ h

0

(1− w2y) dy = 0;

(iv) For all a, b ∈ 0, 1, 2,

(6.1) |∂ax∂byw| ≤ c∗(h

δl

)a(1

δ

)b(hδ)1/2.

Define

(6.2) v(x, y) :=1

2

∫ y

−h(1− w2

y) dy, u(x, y) :=1

2x−

∫ y

−h(wxwy + vx) dy.

Then v = 0 for y = ±h and u = 12x on ∂ω, and v(x,−y) = −v(x, y).

Assume additionally either

(v) w(x, y) = 0 whenever |y| > δ/c∗

or

(vi) δ ≤ h/2 and w(x, y) = α(x)[ψδ(y + ϕ(x)) + ψδ(y − ϕ(x))], where ψδ is an even function

supported on the interval [−δ, δ] satisfying |ψ(k)δ | ≤ c∗/δ

k for k ∈ 0, 1, 2, and where ϕ :[0, l]→ [0, h/2] and |ϕ(k)| ≤ c∗h/lk for k ∈ 0, 1, 2.

Then

(6.3) I(σ,γ)[u , w, ω] ≤ cγδl + c(σl1)2 lh

δ2+ c

h6

δl3

where the constant c may depend only on c∗.

We remark that case (v) corresponds to a single fold which is moving (fold transport), whereas(vi) corresponds to the regions in which one fold is subdivided into two smaller ones (fold splitting).

Remark 6.2 (Assumptions of Lemma 6.1). Recall that the stretching energy of the film wasgiven in (5.1). The definition of u in (6.2) ensures that the second term of the stretching energyvanishes, and the definition of v ensures that the third term vanishes.

15

Proof. First we prove the boundary conditions on v. The condition v(x,−h) = 0 is clear from thedefinition of v. Assumptions (ii) and (iii) imply

(6.4)

∫ h

−h(1− w2

y) dy = 0

hence v(x, h) = 0. Further, by (ii) we get v(x, 0) = 0. Differentiating (6.4) with respect to x gives

(6.5)

∫ h

−hwywxy dy = 0.

Next we show that v is odd in y: By assumption (ii) and equation (6.4)

v(x,−y) =1

2

∫ −y−h

(1− w2y(x, y)) dy =

1

2

∫ h

y

(1− w2y(x, y)) dy

=1

2

∫ h

−h(1− w2

y(x, y)) dy − 1

2

∫ y

−h(1− w2

y(x, y)) dy

= −v(x, y)

as required.Now we turn to the boundary conditions on u. Clearly u = 1

2x for y = −h by the definition ofu. Note that, by (ii), wxwy is an odd function of y. Therefore

(6.6)

∫ h

−hwxwy dy = 0.

Since v is an odd function of y, so is vx, and

(6.7)

∫ h

−hvx dy = 0.

For future use we differentiate this equation with respect to x and record that

(6.8)

∫ h

−hvxx dy = 0.

By equations (6.2)2, (6.6), (6.7)

u(x, h) =1

2x−

∫ h

−h(wxwy + vx) dy =

1

2x.

It remains to show that u = 12x for x = 0, l. This follows immediately from the definition of u and

v and the fact that wx is zero close to x = 0, l (assumption (i)).Now we compute the energy bound (6.3). It is easy to see that the bonding energy IBo :=

γ|(x, y) ∈ ω : w(x, y) > 0| satisfies IBo ≤ cγδl in both cases (v) and (vi), which gives the firstterm in equation (6.3).

To estimate the bending energy we observe that by (iv) and the assumption h ≤ l

|D2w|2 = w2xx + 2w2

xy + w2yy ≤ chδ

((h

δl

)4

+ 2

(h

δl

)2(1

δ

)2

+

(1

δ

)4)≤ chδ

δ4= c

h

δ3.

Therefore in both cases (v) and (vi) the bending energy satisfies

(6.9) (σl1)2

∫ω

|D2w|2 dx ≤ c(σl1)2δlh

δ3= c(σl1)2 lh

δ2,

16

which gives the second term in equation (6.3).Now we come to the stretching energy. By the definition of u and v we only need to estimate

IS =

∫ω

(2ux + w2x − 1)2 dx

(see Remark 6.2). From the definition of u, equation (6.8), and the fact that wywxx is an oddfunction of y (assumption (ii)), we compute

(6.10) ux =1

2−∫ y

−h(wxywx + wywxx + vxx) dy =

1

2+

∫ h

y

(wxywx + wywxx + vxx) dy.

From the definition of v and equation (6.5) we see that

(6.11) vx = −∫ y

−hwywxy dy =

∫ h

y

wywxy dy

and therefore

(6.12) vxx = −∫ y

−h(w2

xy + wywxxy) dy =

∫ h

y

(w2xy + wywxxy) dy.

By assumption (iv)(6.13)

|wxywx + wywxx| ≤(h

δl

)(1

δ

)(hδ)1/2

(h

δl

)(hδ)1/2 +

(1

δ

)(hδ)1/2

(h

δl

)2

(hδ)1/2 ≤ c h3

δ2l2.

First we consider case (v). Since w = 0 for |y| ≥ δ, then from equations (6.10) and (6.12) wesee that ux = 1

2 for |y| ≥ δ. Therefore the stretching energy reduces to

(6.14) IS =

∫ l

0

∫ δ

−δ(2ux + w2

x − 1)2 dydx.

For |y| < δ equations (6.10), (6.12) reduce to

ux =1

2−∫ y

−δ(wxywx + wywxx + vxx) dy =

1

2+

∫ δ

y

(wxywx + wywxx + vxx) dy,(6.15)

vxx = −∫ y

−δ(w2

xy + wywxxy) dy =

∫ δ

y

(w2xy + wywxxy) dy.(6.16)

By equation (6.16) and assumption (iv)

(6.17) |vxx| ≤ cδ

((h

δl

)2(1

δ

)2

hδ +(hδ)1/2

δ

((h

δl

)2(1

δ

)(hδ)1/2

))≤ c h

3

δ2l2.

Using (6.15), (6.13), (6.17) we estimate

(6.18) |2ux − 1| ≤ cδ(h3

δ2l2+

h3

δ2l2

)≤ c h

3

δl2.

We conclude from equation (6.14), (6.18) and assumption (iv) that

(6.19) IS ≤ 2

∫ l

0

∫ δ

−δ(|2ux − 1|2 + |wx|4) dy dx ≤ clδ h

6

δ2l4= c

h6

δl3.

This gives the third term in equation (6.3) and completes the proof for case (v).

17

We finish by computing the stretching energy for case (vi). We decompose ω as ω = ω1∪ω2∪ω3

where

ω1 := (x, y) ∈ ω : δ − ϕ(x) < y < −δ + ϕ(x),ω2 := (x, y) ∈ ω : −δ + ϕ(x) ≤ y ≤ δ + ϕ(x) ∪ (x, y) ∈ ω : −δ − ϕ(x) ≤ y ≤ δ − ϕ(x),ω3 := (x, y) ∈ ω : y > δ + ϕ(x) ∪ (x, y) ∈ ω : y < −δ − ϕ(x).

Observe that w = 0 in ω1 and ω3. The set ω1 is the central region between the two folds, whichoccupy region ω2. The same arguments used in the proof of case (v) show that the stretchingenergy in region ω3 is zero and that the stretching energy in region ω2 satisfies estimate (6.19).(The integral over (−δ, δ) in (6.19) is replaced with integrals over (−δ − ϕ(x), δ − ϕ(x)) and(−δ+ϕ(x), δ+ϕ(x)).) It remains to compute the stretching energy in the central region ω1. Sincew = 0 in this region we only need to estimate∫

ω1

|2ux − 1|2 dx dy.

First we show that vx = 0 in ω1. We only need to consider values of x such that ϕ(x) > δ. Sincew = 0 in ω1 and ω3, equation (6.11) reduces to the following in ω1:

(6.20) vx = −∫ −ϕ(x)+δ

−ϕ(x)−δwywxy dy =

∫ ϕ(x)+δ

ϕ(x)−δwywxy dy in ω1.

But wywxy is even in y. Therefore

(6.21)

∫ ϕ(x)+δ

ϕ(x)−δwywxy dy =

∫ −ϕ(x)+δ

−ϕ(x)−δwywxy dy.

Combining (6.20) and (6.21) proves that vx = 0 in ω1, as claimed. Recalling (6.2), it follows that

uy = −wxwy − vx = 0 in ω1.

Therefore u(x, y) = u(x, 0) for all (x, y) ∈ ω1. Using assumption (vi) we compute, for y < 0,

wx = α′(x)ψδ(y + ϕ(x)) + α(x)ϕ′(x)ψ′δ(y + ϕ(x)),

wy = α(x)ψ′δ(y + ϕ(x))

(since (x, y) ∈ ω1, we have ϕ(x) ≥ δ). It follows that, again for y < 0,

wxwy = α(x)α′(x)1

2

d

dyψ2δ (y + ϕ(x)) + w2

y(x, y)ϕ′(x).

Integrating gives∫ 0

−hwxwy dy = =

1

2α(x)α′(x)ψ2

δ (ϕ(x))− ψ2δ (−h+ ϕ(x))+ ϕ′(x)

∫ 0

−hw2y dy

=1

2α(x)α′(x)ψ2

δ (ϕ(x)) + ϕ′(x)h

by assumption (iii) and since ϕ(x) ∈ [0, h/2], δ < h/2, and ψδ is supported on [−δ, δ]. Therefore

2u(x, 0)− x = −2

∫ 0

−h(wxwy + vx) dy = −α(x)α′(x)ψ2

δ (ϕ(x))− 2hϕ′(x)− 2

∫ 0

−hvx dy.

If (x, 0) ∈ ω1, then ϕ(x) > δ and so ψδ(ϕ(x)) = 0. Hence

2u(x, y)− x = 2u(x, 0)− x = −2hϕ′(x)− 2∂

∂x

∫ 0

−hv dy for all (x, y) ∈ ω1.

18

We compute the integral on the right-hand side:

−2∂

∂x

∫ 0

−hv dy =

∂

∂x

∫ 0

−h

∫ y

−hw2y(x, y) dy dy =

∂

∂x

∫ 0

−h

∫ 0

y

w2y(x, y) dy dy

by changing the order of integration. Therefore

−2∂

∂x

∫ 0

−hv dy = − ∂

∂x

∫ 0

−hyw2

y(x, y) dy =∂

∂x

∫ h

0

yw2y(x, y) dy.

If (x, 0) ∈ ω1, then

−2∂

∂x

∫ 0

−hv dy =

∂

∂x

∫ ϕ(x)+δ

ϕ(x)−δyw2

y(x, y) dy

=∂

∂x

∫ ϕ(x)+δ

ϕ(x)−δ(y − ϕ(x))w2

y(x, y) dy + hϕ′(x) (by assumption (iii))

=∂

∂x

∫ ϕ(x)+δ

ϕ(x)−δ(y − ϕ(x))α2(x)[ψ′δ(y − ϕ(x))]2 dy + hϕ′(x)

=∂

∂x

∫ δ

−δyα2(x)[ψ′δ(y)]2 dy + hϕ′(x) (by changing variables)

= 2α(x)α′(x)

∫ δ

−δy[ψ′δ(y)]2 dy + hϕ′(x)

= hϕ′(x)

since y[ψ′δ(y)]2 is odd. Therefore, if (x, y) ∈ ω1,

2u(x, y)− x = −hϕ′(x).

We conclude that∫ω1

|2ux − 1|2 dx dy =

∫ω1

h2|ϕ′′(x)|2 dx dy ≤ chl h2(h/l2)2 = ch5

l3≤ c h

6

δl3

since δ < h. Therefore the stretching energy in region ω1 is the same order (or less) than thestretching energy in region ω2 and we have finally arrived at the desired estimate (6.3).

In the following lemma we construct a deformation that takes one fold of width 2δ at x = land splits it into two folds of width 2δ separated by a distance of h − 2δ at x = 0. The film isbonded to the substrate between the two folds. See Figure 6, left.

Lemma 6.3 (Fold splitting construction). Let 0 < δ ≤ h/2. Let ψ ∈ C∞c ((−1, 1); [0, 1]) be an evenfunction with ψ(0) = 1. Define ψδ(y) = ψ(y/δ). Let ϕ ∈ C∞([0, l]; [0, h/2]) satisfy ϕ(0) = h/2,ϕ(l) = 0, ϕ′ = ϕ′′ = 0 in a neighbourhood of x = 0, l, and |ϕ(k)| ≤ c∗h/l

k for k ∈ 0, 1, 2 forsome c∗ > 0. Set

w(x, y) := ψδ(y + ϕ(x)) + ψδ(y − ϕ(x)).

Define α : [0, l]→ (0,∞) by

(6.22) α2(x)

∫ h

0

w2y(x, y) dy = h.

Then w : [0, l]× [−h, h]→ [0,∞) defined by

(6.23) w(x, y) := α(x)w(x, y)

satisfies the assumptions of Lemma 6.1.

19

Proof. Observe that w satisfies assumption (ii) of Lemma 6.1 since ψδ is even. Equation (6.22)ensures that w satisfies assumption (iii). Clearly w also satisfies assumption (vi). It remains tocheck assumptions (i) and (iv).

Since δ ≤ h/2 and |ϕ| ≤ h/2, then w(x, y) = 0 for y in a neighbourhood of ±h. We differentiate(6.22) with respect to x and rearrange to get

(6.24) α′(x) =

−α(x)

∫ h

0

wywxy dy∫ h

0

w2y dy

.

By assumption ϕ′ vanishes in a neighbourhood of x = 0, l, and consequently so do wxy, α′ and

wx(x, y) = α′(x)w(x, y) + α(x)(ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x)))ϕ′(x).

Therefore assumption (i) of Lemma 6.1 is satisfied.Finally we check assumption (iv). Observe that

∫ h

0

w2y(x, y) dy ≥

∫ ϕ(x)+δ

ϕ(x)

w2y(x, y) dy ≥ 1

δ

(∫ ϕ(x)+δ

ϕ(x)

wy(x, y) dy

)2

=1

δw2(x, ϕ(x)) ≥ 1

δ

since ψ(0) = 1. Observing that by definition |wy| ≤ c/δ we conclude

(6.25)1

δ≤∫ h

0

w2y dy ≤

c

δand c(hδ)1/2 ≤ α ≤ c(hδ)1/2.

Since |wy| ≤ cδ−1 and |wxy| ≤ c|ϕ′|δ−2, then (6.24) and (6.25) give

|α′| = |α|

∣∣∣∣∣∫ δ+ϕ(x)

max0,−δ+ϕ(x)wywxy dy

∣∣∣∣∣∣∣∣∣∣∫ h

0

w2y dy

∣∣∣∣∣−1

≤ c |α||ϕ′|

δ≤ c|α| h

δl.

Note that

|wxxy| ≤ |ψ′′′δ (y + ϕ(x)) + ψ′′′δ (y − ϕ(x))||ϕ′|2 + |ψ′′δ (y + ϕ(x)) + ψ′′δ (y − ϕ(x))||ϕ′′|

≤ c 1

δ3

(h

l

)2

+ c1

δ2

h

l2≤ c h

2

δ3l2

since δ < h. Therefore

|α′′| ≤

∣∣∣∣∣−α′∫ h

0

wywxy dy − α∫ h

0

(w2xy + wywxxy) dy

∣∣∣∣∣∫ h

0

w2y dy

+ 2|α|

(∫ h

0

wywxy dy

)2

(∫ h

0

w2y dy

)2

≤ cδ(|α′|δδ−1|ϕ′|δ−2 + |α|δ(|ϕ′|2δ−4 + δ−1h2δ−3l−2)

)+ c|α|δ2

(δδ−1|ϕ′|δ−2

)2≤ c|α|

(h

δl

)2

20

since |α′| ≤ c|α||ϕ′|/δ and |ϕ′| ≤ ch/l. Putting everything together gives

|w| ≤ |α||w| ≤ c(hδ)1/2,

|wx| ≤ |α′||w|+ |α||ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x))||ϕ′| ≤ c|α||ϕ′|δ−1 ≤ c(hδ)1/2 h

δl,

|wy| ≤ |α||ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x))| ≤ c(hδ)1/2 1

δ,

|wxx| ≤ |α′′||w|+ 2|α′||ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x))||ϕ′|+ |α||ψ′′δ (y + ϕ(x)) + ψ′′δ (y − ϕ(x))||ϕ′|2

+ |α||ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x))||ϕ′′|

≤ c(hδ)1/2

(h2

δ2l2+

h

δl2

)≤ c(hδ)1/2

(h

δl

)2

,

|wxy| ≤ |α′||ψ′δ(y + ϕ(x)) + ψ′δ(y − ϕ(x))|+ |α||ψ′′δ (y + ϕ(x)) + ψ′′δ (y − ϕ(x))||ϕ′|

≤ c(hδ)1/2

(h

δl

)(1

δ

),

|wyy| ≤ |α||ψ′′δ (y + ϕ(x)) + ψ′′δ (y − ϕ(x))| ≤ c(hδ)1/2

(1

δ

)2

and hence w satisfies assumption (iv) of Lemma 6.1.

In the following lemma we construct a deformation that takes one fold of width 2δ at x = land shrinks it down to a fold of width 2δ < 2δ at x = 0. See Figure 6, right.

Lemma 6.4 (Fold shrinkage construction). Let 0 < δ ≤ h, λ ∈ [1/4, 1]. Let ψ ∈ C∞c ((−1, 1); [0, 1])

be an even function with ψ(0) = 1. Define ψδ(y) := ψ(y/δ). Let δ ∈ C2([0, l]; [λδ, δ]) satisfy δ(0) =

λδ, δ(l) = δ, δ′ = 0 in a neighbourhood of x = 0, l, and |δ(k)| ≤ cδ/lk. Set w(x, y) := ψδ(x)(y).

Define α : [0, l]→ (0,∞) by

(6.26) α2(x)

∫ h

0

w2y(x, y) dy = h.

Then w : [0, l]× [−h, h]→ [0,∞) defined by

(6.27) w(x, y) := α(x)w(x, y)

satisfies the assumptions of Lemma 6.1.

Proof. The same arguments used in the proof of Lemma 6.3 show that w satisfies assumptions(i)–(iii) of Lemma 6.1. Clearly it also satisfies assumption (v). It remains to check assumption(iv). Observe that∫ h

0

w2y(x, y) dy =

∫ h

0

[ψ′δ(x)

(y)]2 dy =1

δ2(x)

∫ δ(x)

0

[ψ′(

y

δ(x)

)]2dy =

1

δ(x)

∫ 1

0

[ψ′(y)]2 dy =c∗

δ(x).

Therefore

α(x) =

(δ(x)h

c∗

)1/2

≤ c(δh)1/2.

Note that∂

∂δψδ(y) = −ψ′

(yδ

) y

δ2.

21

Since δ(x) ≥ λδ, we estimate

|wx(x, y)| =

∣∣∣∣∣ψ′(

y

δ(x)

)y

δ2(x)δ′(x)

∣∣∣∣∣ ≤ c∣∣∣∣∣ δ′(x)

δ(x)

∣∣∣∣∣ ≤ c

l,

|wy(x, y)| =

∣∣∣∣∣ψ′(

y

δ(x)

)1

δ(x)

∣∣∣∣∣ ≤ c

δ,

|wyx(x, y)| ≤

∣∣∣∣∣ψ′′(

y

δ(x)

)y

δ3(x)δ′(x)

∣∣∣∣∣+

∣∣∣∣∣ψ′(

y

δ(x)

)δ′(x)

δ2(x)

∣∣∣∣∣ ≤ c∣∣∣∣∣ δ′(x)

δ2(x)

∣∣∣∣∣ ≤ c

δl,

|wxxy(x, y)| ≤

∣∣∣∣∣ψ′′′(

y

δ(x)

)y2

δ5(x)(δ′(x))2

∣∣∣∣∣+ 3

∣∣∣∣∣ψ′′(

y

δ(x)

)y

δ4(x)(δ′(x))2

∣∣∣∣∣+

∣∣∣∣∣ψ′′(

y

δ(x)

)y

δ3(x)δ′′(x)

∣∣∣∣∣+

∣∣∣∣∣ψ′(

y

δ(x)

)1

δ2(x)δ′′(x)

∣∣∣∣∣+ 2

∣∣∣∣∣ψ′(

y

δ(x)

)1

δ3(x)(δ′(x))2

∣∣∣∣∣≤ c 1

δl2.

Similarly to the proof of Lemma 6.3, it follows that

|α′| ≤ c |α|l, |α′′| ≤ c |α|

l2.

The estimates on the derivatives of w and α are at least as good as those given in the proof ofLemma 6.3 (in fact they are better). Therefore the derivatives of w satisfy the same estimatesproved in Lemma 6.3, and w satisfies assumption (iv) of Lemma 6.1, as required.

Remark 6.5. From the proof of Lemma 6.4 we see that fold shrinkage is cheaper than foldsplitting.

6.2 Overall construction.

In this section we put everything together to prove Theorem 2.1 for the branching regimes Cand D.

Proposition 6.6 (Overall construction for 0 ≤ γ ≤ σ−4/9). Let 0 < l1 ≤ l2 and γ ≤ σ−4/9. Then

minV

I(σ,γ) ≤ cl1l2(σ1/2γ5/8 + σ).

Remark 6.7. Observe that

σ1/2γ5/8 + σ ≤

2σ1/2γ5/8 if γ ≥ σ4/5, in particular if (σ, γ) ∈ C,2σ if γ < σ4/5, i.e., if (σ, γ) ∈ D.

Alsoσ1/2γ5/8 + σ ≥ (σγ)2/5 if γ ≥ σ−4/9, in particular if (σ, γ) ∈ B.

Proof of Proposition 6.6. We cover Ω = (0, l1) × (0, l2) by the union of three sets: a boundarylayer region ΩBL, a branching region ΩBR, and a bulk region ΩB,

Ω ⊂ Ω := ΩBL ∪ ΩBR ∪ ΩB

where

ΩBL := (0, ε)× (0, l2), ΩBR := [ε, ε+ lBR]× (0, l2), ΩB := (ε+ lBR, ε+ lBR + l1)× (0, l2),

22

x

y

l1

l2

ε l1

lBR

2L0

2L1

ΩBΩBL Ω1

V 30U3

0

Figure 8: Sketch of the geometry in the proof of Proposition 6.6. The sets are defined so that Ω is covered bytheir union, this is ensured by making ΩB have width l1.

and where the boundary layer width ε and the branching region width lBR are to be determined.In particular, lBR will be chosen by summing up the optimal widths of the individual branchingsteps. We construct the test function on Ω and then restrict to Ω, which only decreases the energy;note that Ω ⊂ Ω for any choice of the parameters. In the boundary layer region ΩBL we use theconstruction (u, v, w) from Section 5.2 with h = hN and δ = δN to be determined. In the bulkregion ΩB we use the construction from Section 5.1 with l = l1, h = h0 and δ = δ0, again to bedetermined. We assume hN = 2Nh0 for some N ∈ N, also to be determined. (We assume withoutloss of generality as in Section 5.3 that 2h0 divides l2 and use the construction from Section 5.1on the l2/2h0 rectangles (ε+ lBR, ε+ lBR + l1)× (2h0(j − 1), 2h0j), j ∈ 1, . . . , l2/2h0.)

We split up the branching region ΩBR into N vertical strips Ωi of width 2Li, i ∈ 0, . . . , N−1:

ΩBR =

N−1⋃i=0

Ωi, Ωi :=

[0, 2Li] + ε+

N−1∑k=i+1

2Lk

× (0, l2).

Note that the strips are labelled from left to right in decreasing order ΩN−1, . . . ,Ω1,Ω0. The total

width of the branching region is defined by lBR :=∑N−1i=0 2Li, while the values of the Li’s are

chosen below. Let hi+1 = hi/2, i ∈ 0, . . . , N − 1. Then each strip is split into l2/2hi rectanglesRji of height 2hi, see Figure 8.

Ωi =

l2/2hi⋃j=1

Rji , Rji :=

[0, 2Li] + ε+

N−1∑k=i+1

2Lk

×

[0, 2hi] + (j − 1)2hi

.

Finally, we split each rectangle Rji into two rectangles U ji , V ji of width Li:

Rji = U ji ∪ Vji

U ji :=

[0, Li] + ε+

N−1∑k=i+1

2Lk

×

[0, 2hi] + (j − 1)2hi

,

V ji :=

[Li, 2Li] + ε+

N−1∑k=i+1

2Lk

×

[0, 2hi] + (j − 1)2hi

.

23

Ui Vi

splitting shrinkage

Vi+1Ui+1

LiLi+1

hi = 2hi+1

2hi

2δi+1

2δi

Figure 9: Sketch of the geometry in the proof of Proposition 6.6. The area marked green is the set wherew > 0.

Let δ0 > δ1 > · · · > δN . On the rectangles U ji we use the fold splitting construction from Lemma6.3 with h = hi, δ = δi+1, l = Li. This is admissible provided that

(6.28) δi+1 ≤ hi+1 =hi2

and hi ≤ Li .

On the rectangles V ji we use the fold shrinkage construction from Lemma 6.4 with h = hi, δ = δi,λ = δi+1/δi, l = Li, see Figure 9. This is admissible provided that

(6.29) δi ≤ hi ≤ Li and1

4δi ≤ δi+1 ≤ δi .

By Lemmas 6.1, 6.3 and 6.4 the energy of the displacement satisfies

(6.30) I(σ,γ)[u , w, U ji ∪ Vji ] ≤ cγδiLi + c(σl1)2Lihi

δ2i

+ ch6i

δiL3i

.

Equating terms on the right-hand leads to the optimality conditions

δi ∼ (σl1)2/3γ−1/3h1/3i , Li ∼ (σl1)−1/2δ

1/4i h

5/4i .

In order to enforce (6.28) and (6.29) we choose

δi := minhi, (σl1)2/3γ−1/3h

1/3i

, Li := max

hi, (σl1)−1/2δ

1/4i h

5/4i

, i ∈ 0, . . . , N − 1.

We only have to verify the second condition in (6.29). Obviously δi+1 ≤ δi. At the same time,hi+1 = hi/2 implies δi+1 ≥ δi/2.

To complete the proof we need to consider two cases: γ ≥ 1, γ < 1.Case γ ≥ 1: First we choose hN and δN , the size of the folding pattern in the boundary layer,

by equating terms in equation (5.6). This gives

(6.31) ε := hN := σl1γ, δN := σl1

and

(6.32) I(σ,γ)[u , w,ΩBL] ≤ cl2ε = cl1l2σγ.

Observe that δN ≤ ε since γ ≥ 1 and so the bound (5.6) is valid (in particular, h ≤ l2 sinceσγ ≤ 1). Now we determine which of the two values is taken by δi, i ∈ 0, . . . , N − 1. Note that

hi ≥ (σl1)2/3γ−1/3h1/3i ⇐⇒ hi ≥ σl1γ−1/2.

But for all ihi ≥ hN = σl1γ ≥ σl1γ−1/2

since γ ≥ 1. Therefore for all i ∈ 0, . . . , N − 1

(6.33) δi = (σl1)2/3γ−1/3h1/3i

24

and, inserting in the definition of Li,

Li = maxhi, (σl1)−1/3γ−1/12h

4/3i

.

Note thathi ≤ (σl1)−1/3γ−1/12h

4/3i ⇐⇒ hi ≥ σl1γ1/4.

But for all ihi ≥ hN = σl1γ ≥ σl1γ1/4

since γ ≥ 1. Therefore

(6.34) Li = (σl1)−1/3γ−1/12h4/3i for all i ∈ 0, . . . , N − 1.

It remains to choose h0 (equivalently N). We do this by balancing the energy in the bulk regionΩB with the energy in the branching region ΩBR. The same computation as in Lemma 5.1 showsthat the energy in ΩB satisfies

(6.35) I(σ,γ)[u , w,ΩB] ≤ c l22h0

((σl1)2 l1h0

δ20

+ 2γl1δ0

)= cl1l2(σl1)2/3γ2/3h

−2/30 .

Inserting (6.33) and (6.34) in (6.30) leads to

I(σ,γ)[u , w, U ji ∪ Vji ] ≤ c(σl1)1/3γ7/12h

5/3i .

Therefore the energy in ΩBR satisfies

I(σ,γ)[u , w,ΩBR] ≤ cN−1∑i=0

l22hi

(σl1)1/3γ7/12h5/3i = c

N−1∑i=0

l2(σl1)1/3γ7/12h2/3i

= c

N−1∑i=0

l2(σl1)1/3γ7/12(2−ih0)2/3 ≤ cl2(σl1)1/3γ7/12h2/30 .(6.36)

Equating the energy bounds for ΩB and ΩBR gives

l1l2(σl1)2/3γ2/3h−2/30 = l2(σl1)1/3γ7/12h

2/30 ⇐⇒ h0 = l1σ

1/4γ1/16.

(Here we have assumed that the equation l1σγ = hN = 2−Nh0 = 2−N l1σ1/4γ1/16 has an integer

solution N . In general this will not be the case. We obtain the same energy bound, however, bydefining h0 = l1σ

1/4γ1/16, N = b| log2(l1σγ/h0)|c, hi = 2−ih0, i ∈ 1, . . . , N.) We remark thath0 ≥ hN because γ ≤ σ−4/5. Substituting h0 = l1σ

1/4γ1/16 into (6.35) and (6.36) yields

I(σ,γ)[u , w,ΩB ∪ ΩBR] ≤ cl1l2σ1/2γ5/8.

By combining this and (6.32) we find that

I(σ,γ)[u , w,Ω] ≤ cl1l2(σ1/2γ5/8 + σγ).

But since σγ ≤ 1 and γ ≥ 1

σγ = σ1/2γ5/8(σγ)1/2γ−1/8 ≤ σ1/2γ5/8

and we obtain the desired result.Case γ < 1: In this case we choose ε, hN and δN as for the case γ = 0 (see [10]):

(6.37) ε = hN = δN = l1σ.

25

(Compare (6.37) to (6.31).) Substituting these into equation (5.6) gives

(6.38) I(σ,γ)[u , w,ΩBL] ≤ cl1l2σ(1 + γ) ≤ cl1l2σ

since γ < 1. Now we determine δi for i ∈ 0, . . . , N − 1. In this case

hN = σl1 < σl1γ−1/2.

Let I be the largest value of i < N such that hi ≥ σl1γ−1/2 (if there is none, I = −1). We have

(6.39) δi = (σl1)2/3γ−1/3h1/3i for i ∈ 0, . . . , I, δi = hi for i ∈ I + 1, . . . , N − 1.

Therefore

Li =

maxhi, (σl1)−1/3γ−1/12h

4/3i

if i ∈ 0, . . . , I,

maxhi, (σl1)−1/2h

3/2i

if i ∈ I + 1, . . . , N − 1.

Since γ < 1 and hi ≥ hN = σl1, it is easy to check that

(6.40) Li =

(σl1)−1/3γ−1/12h

4/3i if i ∈ 0, . . . , I,

(σl1)−1/2h3/2i if i ∈ I + 1, . . . , N − 1.

It remains to choose h0. We do this as for the case γ ≥ 1 by balancing the energy in the bulkregion ΩB with the energy in the branching region ΩBR. From (6.35) we see that the energy inthe bulk region satisfies

(6.41) I(σ,γ)[u , w,ΩB] ≤

cl1l2(σl1)2/3γ2/3h

−2/30 if h0 ≥ σl1γ−1/2,

cl1l2(σl1)2h−20 if h0 < σl1γ

−1/2.

Inserting equations (6.39), (6.40) in (6.30),

I(σ,γ)[u , w, U ji ∪ Vji ] ≤ c

(σl1)1/3γ7/12h

5/3i if i ∈ 0, . . . , I,

(σl1)−1/2γh5/2i + (σl1)3/2h

1/2i if i ∈ I + 1, . . . , N − 1.

Therefore the energy in ΩBR satisfies

I(σ,γ)[u , w,ΩBR] ≤ cI∑i=0

l22hi

(σl1)1/3γ7/12h5/3i + c

N−1∑i=I+1

l22hi

((σl1)−1/2γh

5/2i + (σl1)3/2h

1/2i

)≤ cl2

((σl1)1/3γ7/12h

2/30 + (σl1)−1/2γh

3/2I+1 + (σl1)3/2h

−1/2N−1

)≤ cl2

((σl1)1/3γ7/12h

2/30 + σl1γ

1/4 + σl1

)since hI−1 < σl1γ

−1/2 and hN−1 = 2hN = 2σl1. Since γ < 1,

(6.42) I(σ,γ)[u , w,ΩBR] ≤ cl2(

(σl1)1/3γ7/12h2/30 + σl1

).

It remains to choose h0. If γ < σ, we expect the second term on the right-hand side of (6.42) willdominate the energy, therefore it suffices to choose h0 so that the others are not bigger. Focusingon the second row of (6.41) one easily sees that h0 = σ1/2l1 is the appropriate choice (this isthe same choice used in [10, 11]). In particular, since γ < σ this obeys h0 < σl1γ

−1/2, hence isconsistent. Adding together (6.41), (6.38) and (6.42) yields in this case

I(σ,γ)[u , w,Ω] ≤ cl1l2(σ + σ2/3γ7/12) ≤ 2cl1l2σ.

26

If instead γ ≥ σ, we choose h0 by equating the h0 term on the right-hand side of (6.42) with thefirst row in the right-hand side of (6.41). This gives

h0 = l1σ1/4γ1/16

as in the case γ ≥ 1. This is consistent since h0 ≥ σl1γ−1/2 is the same as γ ≥ σ4/3, which is truein this case. Then adding together (6.41), (6.38) and (6.42) yields

I(σ,γ)[u , w,Ω] ≤ cl1l2(σ + σ1/2γ5/8)

as required.

A Derivation of the Model and Rescaling

In this appendix we derive the energy (1.1). We model the two-layer material (an elastic film ona substrate) described in the introduction with an energy consisting of two parts, an elastic energyfor the thin film and a bonding energy for the interaction between the film and the substrate. Wetake the elastic energy to be the von Karman energy, which penalises extension (stretching andcompression) and bending:

(A.1)

IvK :=Ehf

2(1− ν2)

∫Ω

(1− ν)

∣∣∣∣Du +DuT

2+Dw ⊗Dw

2− ε∗Id

∣∣∣∣2

+ ν

∣∣∣∣tr(Du +DuT

2+Dw ⊗Dw

2− ε∗Id

)∣∣∣∣2dx

+Eh3

f

24(1− ν2)

∫Ω

(1− ν)|D2w|2 + ν(∆w)2

dx

where Ω = (0, l1)×(0, l2) is the set of material points (x, y) of the film, hf is the thickness of the film,E is the Young’s Modulus, and ν is the Poisson ratio. The functions u(x, y) = (u(x, y), v(x, y)) andw(x, y) are the in-plane and transversal (vertical) displacements of the film from the isotropicallycompressed state ((1 − ε∗)x, (1 − ε∗)y, 0), where 0 < ε∗ < 1 is the compression ratio. Notethat ε∗ = 0 corresponds to no compression and ε∗ = 1 corresponds to total compression. Thus(u, v, w) = (0, 0, 0) corresponds to the isotropically compressed state of the film and (u, v, w) =(ε∗x, ε∗y, 0) corresponds to the relaxed, natural state. The substrate is taken to be at heightz = 0. Therefore the transversal displacement w must satisfy the constraint w ≥ 0 (since the filmcannot go below the substrate). If w(x, y) = 0 then the material point (x, y) of the film is bondeddown to the substrate.

For the energy scaling analysis that we perform we can set the Poisson ratio ν equal to zerowithout loss of generality since the terms of (A.1) with value of ν in the physical range (−1, 1/2)can be bound from above and below by those without, as shown in [10, Appendix B].

The interfacial force between the thin film and the substrate is an attractive van der Waalsforce. We model it in a simple way with a bonding energy IBo that penalises debonding from thesubstrate:

(A.2) IBo := γ∗|(x, y) ∈ Ω : w(x, y) > 0|

where γ∗ is a constant depending on the material properties of the film and the substrate. Inmore sophisticated models IBo would also depend on the size of w, i.e., how far the film is fromthe substrate.

Thus the total energy we assign to the system is

(A.3) I := IvK + IBo.

27

On the boundary x = 0 we assume that the film is fixed to the substrate and satisfies theclamped boundary conditions

(A.4) u(0, y) = 0, v(0, y) = 0, w(0, y) = 0, Dw(0, y) = 0.

(Note that in the experiments of [31] the film is actually fixed to a buffer layer and satisfies slightlydifferent boundary conditions. This is discussed in Appendix B.) The film is free on the rest ofits boundary, x = l1 ∪ y = 0 ∪ y = l2.

Rescaling. In order to reduce the number of parameters we define rescaled variables, denotedwith a superscript ∗, by

u =: 2ε∗u∗, w =: (2ε∗)

1/2 w∗.(A.5)

By substituting from equation (A.5) into equation (A.3), multiplying by 2(1−ν2)/(Ehfε2∗), setting

ν = 0, and dropping all the superscripts * for convenience, we obtain a new energy I(σ,γ):

(A.6)

I(σ,γ)[u , w, l1, l2] =

∫ l1

x=0

∫ l2

y=0

|Du +DuT +Dw ⊗Dw − Id|2 dx dy

+ (σl1)2

∫ l1

x=0

∫ l2

y=0

|D2w|2 dx dy + γ|(x, y) ∈ Ω : w(x, y) > 0|

where σ is the rescaled film thickness and γ is the rescaled bonding energy per unit area:

(A.7) σ :=hf

l1(6ε∗)1/2, γ :=

2(1− ν2)γ∗

Ehfε2∗

=2γ∗

Ehfε2∗

(since ν = 0).

Throughout this paper we refer to γ as the bonding strength, although note that it also depends onthe unscaled film thickness hf. Equation (A.6) is exactly equation (1.1) given in the Introduction.

B Upper Bounds for Buffer Layers with Nonzero Thickness

In the experiments of [31] the film is not clamped to the substrate, but rather to a thin bufferlayer, and satisfies clamped boundary conditions of the form

(B.1) u(0, y) = 0, v(0, y) = 0, w(0, y) = hb, Dw(0, y) = 0

where hb > 0 is the thickness of the buffer layer. In this paper we considered the unphysical casehb = 0 in order to simplify the analysis. In this appendix we show that the upper bounds ofTheorem 2.1 also hold when hb is sufficiently small. Recall that Ω = (0, l1)× (0, l2). Define

Vhb=

(u , w) ∈ H1(Ω;R2)×H2(Ω) : w = hb, u = Dw = 0, on 0 × (0, l2), w ≥ 0.

Theorem B.1 (Upper Bounds for hb > 0). Let 0 < l1 ≤ l2, σ ∈ (0, 1), γ ≥ 0. Assume that hbsatisfies

0 ≤ hbl1≤ minσ, σ−1γ−3/2.

Then there exists admissible displacement fields (u , w) ∈ Vhbsatisfying the same upper bounds as

in Theorem 2.1.

Proof. We construct displacement fields satisfying the required upper bounds by simply appendingto the constructions used to prove Theorem 2.1 a boundary layer in which the film is bent upwardsfrom height 0 to height hb. Let η > 0 be the boundary layer thickness. For x ∈ [0, η], y ∈ [0, l2]define

(B.2) u =1

2x, v = 0, w = hb

[1− ψ

(x

η

)]

28

where ψ ∈ C2(R) is a function with ψ(t) = 0 for t ≤ 0, ψ(t) = 1 for t ≥ 1. For x ≥ η, define u,v, w as in the proof of Theorem 2.1 (except that u is replaced with u+ η/2). The energy EηBL of(u, v, w) in the boundary layer (0, η)× (0, l2) satisfies

(B.3) EηBL ≤ l2η(h4

b

η4+ (σl1)2h

2b

η4+ 1 + γ

)≤ l2η

(2(σl1)2h

2b

η4+ 1 + γ

)since we assumed that hb ≤ σl1. We choose

(B.4) η = (σl1hb)1/2 max1, γ−1/4

so that

(B.5) EηBL ≤ l2(σl1hb)1/2 max1, γ3/4.

For each regime it is easy to check that this boundary layer energy is no greater than the energyof the constructions given in Theorem 2.1.

This is the simplest possible modification of the proof of Theorem 2.1 and we do not claimthat the bound hb/l1 < minσ, σ−1γ−3/2 is sharp.

C Poincare Constant

In this section we prove the Poincare inequality (3.2). It is sufficient to prove the following:

Lemma C.1. Let Q = (0, 1)× (0, 1) be the unit square and f ∈W 1,2(Q;R2). Let f be zero on atleast half of the square:

|x ∈ Q : f (x ) = 0| ≥ 1

2.

Then ∫Q

|f |2 dx ≤∫Q

|Df |2 dx .

Proof. First we recall the Poincare inequality in one dimension. Let g ∈ C1([0, 1];R2). Write gas the Fourier cosine series (the Fourier series of the even extension of g to [−1, 1])

g(x) =g0

2+

∞∑k=1

gk cos(πkx), gk = 2

∫ 1

0

g(x) cos(πkx) dx.

Let g = g0/2 =∫ 1

0g(x) dx. By Parseval’s Theorem

(C.1)∫ 1

0

|g(x)− g |2 dx =1

2

∞∑k=1

|gk|2 =1

2π2

∞∑k=1

1

k2|πkgk|2 ≤

1

2π2

∞∑k=1

|πkgk|2 =1

π2

∫ 1

0

|g ′(x)|2 dx,

which is the Poincare inequality in one dimension. By density the same holds for g ∈W 1,2((0, 1);R2).We now turn to the two-dimensional case. By Fubini’s theorem, for almost every x one has

f (x, ·) ∈W 1,2((0, 1);R2). Let

a(x) =

∫ 1

0

f (x, y) dy, a =

∫ 1

0

a(x) dx =

∫Q

f (x, y) dxdy.

By equation (C.1)∫ 1

0

|a(x)− a |2 dx ≤ 1

π2

∫ 1

0

|a ′(x)|2 dx ≤ 1

π2

∫ 1

0

∫ 1

0

|fx|2 dxdy,∫ 1

0

|f (x, y)− a(x)|2 dy ≤ 1

π2

∫ 1

0

|fy|2 dy.

29

Therefore

(C.2)

∫Q

|f − a |2 dxdy ≤ 2

(∫Q

|f − a |2 dxdy +

∫Q

|a − a |2 dxdy)≤ 2

π2

∫Q

|Df |2 dxdy.

Let U = x ∈ Q : f (x ) = 0. We have

(C.3)1

2|a |2 ≤ |U ||a |2 =

∫U

|f − a |2 dx ≤ 2

π2

∫Q

|Df |2 dxdy.

Observe that ∫Q

|f − a |2 dx =

∫Q

|f |2 dx − |a |2.

By combining this with estimates (C.2) and (C.3) we obtain∫Q

|f |2 dx =

∫Q

|f − a |2 dx + |a |2 ≤ 6

π2

∫Q

|Df |2 dx

which, since 6 < π2, completes the proof.

Acknowledgements. The authors would like to thank O. G. Schmidt and other members ofthe Institute for Integrative Nanosciences, IFW Dresden (including P. Cendula, S. Kiravittayaand Y. F. Mei) for interesting discussions about the experiments that were the motivation for thispaper. This work was partially supported by the Deutsche Forschungsgemeinschaft through theSonderforschungsbereich 1060 “The mathematics of emergent effects”.

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